We study the propagation of neutrinos in gravitational fields using wave functions that are exact to first order in the metric deviation. For illustrative purposes, the geometrical background is represented by the Lense-Thirring metric. We derive explicit expressions for neutrino deflection, helicity transitions, flavor oscillations and oscillation Hamiltonian.

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This comment is in response to Nieves and Pal [1] who dispute our claim [2] that a classical gravitational field may possibly distinguish between Dirac and Majorana neutrinos, described in terms of Gaussian wave packets propagating in a Lense-Thirring background, where the distinction is manifested in spin-gravity corrections to the neutrino oscillation length. They contend that our model for Majorana neutrinos is incorrect and that any conclusions relating to this neutrino type are not reliable. Furthermore, they suggest that any distinction between the two neutrino types will be suppressed by factors of m/E where m is the neutrino mass and E is its mean energy of propagation, and claim that this distinction is unobservable when m/E ≪ 1. We beg to disagree.As noted above, our model for the Dirac and Majorana neutrinos is motivated by a wave packet approach in quantum mechanics, as opposed to a plane-wave expansion in quantum field theory. This is an essential detail which Nieves and Pal fail to acknowledge. In addition, the gravitational field [2] is incorporated in terms of a gravitational phase Φ G , giving rise to an interaction Hamiltonian H ΦG with spin-dependent features, to be evaluated in terms of time-independent perturbation theory. These two details are important for framing the context underpinning our reply. Regarding the technical concerns, we agree that the Majorana condition they note in their eq. (1) is certainly true for a fermion field operator. However, we again emphasize that our perspective is quantum mechanical, so our treatment of the Majorana condition must be described in terms of wave functions. Adopting their notation, our approach is to identify [3,4] Unlike what Nieves and Pal claim, it indeed follows [3] that W 1,2 is a solution of the free particle equation / k W 1,2 = ± m W 1,2 [5] for wave functions. However, a more precise implementation of (1) with our notation leads to a Majorana wave packet model with the formwhereand ξ(k) is the Gaussian function [2]. Clearly, it follows from (2) that |ψ c 1(2) Maj. = ± |ψ 1(2) Maj. . This leads to a modification of our eq. (13) for the Majorana matrix element [2], which isOur plots [2], as applied to the SN 1987A data, are completely unaffected by the adjustments because the corrections only alter the contributions coupled to M ΩR 2 /r 2 , which are all exponentially damped compared to the M/r contributions. As for the m/E suppression issue raised by Nieves and Pal, this is of no relevance because all such terms are automatically excluded within the construction of |W 1,2 (k) Maj. , so everything presented in [2] is of leading order. As shown in (5), and also present in our eq. (13)

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It is suggested that the rotation-spin coupling predicted by Mashhoon may lead, in the case of neutrinos, to a helicity flip that has implications for astrophysical objects such as rotating neutron stars and supernovae. The coupling is generalized here to include gravitational fields and total angular momentum and is derived by solving the covariant Dirac and Maxwell-Proca equations exactly to first order in the metric deviation y^v. For fermions the spin part of the effect also applies to gravitational fields of arbitrary strength.The solution of wave equations in weak inertial or gravitational fields is usually based on nonrelativistic approximations or the eikonal approximation and usually neglects spin. This may no longer be sufficient to cover all physical situations one meets in particle interferometry or in the description of phenomena by accelerated observers. The point of view adopted in this paper is that of general relativity, which treats both inertial and gravitational phenomena in a unified way. The general characteristic of the solutions of the covariant Klein-Gordon and Landau-Ginzburg equations previously given by us 1 is that they can be written in the form (p^e ~'*0, where O is a general solution of the field-free equations,P represents path ordering and L afi (z),P a are generators of the Poincare group and the path integral is taken along the classical trajectory of the particle. For a closed path, the gravitational phase is related to Berry's phase 2 and can be rewritten as 3 #= -j f S^R^op xL afi dr^v, where X p is the surface bound by the closed path p and R MVa p represents the linearized Reimann curvature tensor. The purpose of this Letter is to further extend the above results to particles with spin and to present first-order solutions of the covariant Dirac and Maxwell equations. These solutions reproduce the spinrotation coupling proposed by Mashhoon 4 and, for Dirac particles, suggest some interesting astrophysical consequences.Spin-^-particles.-Consider the covariant Dirac equationwhere the matrices y^(x) are obtained from the usual Dirac matrices y a by means of a vierbein field h M a (x).In Eq.(2) /> Ai = V /i + iT p , where V^ is the usual covariant derivative. The spinoral connection T^ is given byA new spinor y/ defined by y/ = exp[/ xPfpr^(z)dz^]y/ satisfies the equation [y M (x)\^ -mc/hty-0, or (g*%V v -/w 2 c 2 M 2 V-0. This can be solved in the weak-field approximation for each component of y/' and the gravitational field appears only in the phase factor of Eq. (1). One therefore has y/=exp(--/^)^, where ^ satisfies the Dirac equation in Minkowski space, and finallyFor a closed path /?, S becomes S=P § r M (z)dz»-{ Jf z R fl vafiL aP dT' Iv . (4) By applying Stokes theorem to the first term of S, one obtains S--7 ffz p R,va P C7 afi dT» Vi ffz p R»vapL a PdT»\ with the help of the relationship 5 r^(x)-r^v(jc)+/[r^(x),r w (x)] = -U fl^^.One may therefore introduce the total angular momentum J aP =L af >+a aP and rewrite S as 5--i " ff Lp R»vapJ afi dTK\ ( 5 )Spin-1 particles...

The problem of particle interferometry in weak inertial or gravitational fields is treated from a unified point of view. The effect of the fields can be confined to a phase factor to be determined by quadratures once the solution of the possibly non-linear wave equation describing the particles is known. The procedure is completely Lorentz invariant and gauge invariant, can be extended to higher orders and applies to a wide range of interferometers, from optical ones to those using superfluids. Results already reported in the literature are re-obtained and in some cases improved. Other results known to hold for stationary fields are extended to time-dependent fields. It is also shown that interferometers hold promise as broad-band detectors of gravitational radiation even at high frequencies.

The basis on which Weyl's unified theory of gravitation and electromagnetism was rejected is reconsidered from a new perspective. It is argued that while Weyl's theory, as indeed any classical theory, is incapable of explaining atomic phenomena, this does not nullify the geometric interpretation of the exterior electromagnetic field; it simply reflects the fact that some form of quantization is needed to account for atomic standards of length. In support of this argument the Gauss-MainardiCodazzi formalism is employed to demonstrate that it is possible to construct a bubble in Weyl space where the exterior geometry is conformally invariant and the electromagnetic field can be given a geometric interpretation, while at the same time a standard of length can be introduced into the theory by breaking the conformal invariance in the interior of the bubble.PACS number(s): 04.50.+h, 04.20.C~

We show that, in cosmological microlensing, corrections of order v/c ∼ ∆λ/λ, to the deflection angle of light beams from a distant source are not negligible and that all microlensing quantities should be corrected up to this order independently of the cosmological model used.keywords: microlensing, redshift, cosmology.In the last decade, gravitational lensing has become one of the most powerful tools in astrophysics and cosmology in studies about the presence and mass distribution of dark matter in the Universe [1],[2], [3]. It affords, in principle, estimates of the gravitational mass of all large scale structures, from galaxies to super clusters, and, in the specific application called microlensing, it can be used to search for Massive Astrophysical Compact Halo Objects (MACHOs) [4]. These objects are considered to be the main constituents of the dark halo of spiral galaxies (of our Galaxy, in particular) and, theoretically, could range in mass from 10 −8 ÷ 10 2 M ⊙ . MACHOs could therefore represent planets, brown dwarfs, or massive black holes [5]. The fundamental problem is how lensing by a pointlike mass can be detected. Unless the lens is very massive (M > 10 6 M ⊙ ), the angular separation of two images, usually produced by a point-like lens, is too small to be resolved. The angular separations of images are, in fact, of the order ∼ 10 −3 ÷ 10 −6 arcsec, hence the term microlensing. However, even when detecting multiple images is impossible, the magnification can still be seen if the lens and the source move relative to each other. This motion gives rise to a lensing-induced time dependence of the source luminosity [6]. *